2.4 and10.2 Direct/Inverse/Joint Variation ©2001 by R. Villar All Rights Reserved

Direct, Inverse and Joint Variation Direct Variation is a linear function. For example d = r t If time is constant, as rate goes up, distance goes up. We can say, distance varies directly with rate... Direct variation equations can be written as y = kx where k is the constant of variation. Example: The values of x and y vary directly. Write an equation that relates the variables if x = 3 when y = 12. y = kx 12 = k 3 k = 4 y = 4x

Example: The area A (in square inches) of a rectangle varies directly with its width, W, (in inches). When the width is 4 inches, the area is 12 square inches. Write an equation that relates A and W. Find the width of the rectangle when the area is 24 square inches. W = k A This is the model for inverse variation. 4 = k 12 Substitute to solve for k.. 1/3 = k W = 1/3 A Find W when A is 24. W = 1/3 24 W = 8 inches

Inverse Variation The variables x and y vary inversely if, for a constant k, yx = k or y =k x The number k is the constant of variation. Example: The variables x and y vary inversely. Find an equation that relates the variables if x = 4 when y = –3. yx = k yx = –12

Example: The volume of a gas V at a constant temperature varies inversely with its pressure, P. The pressure acting on 15 m 3 of air is raised from 1 atmosphere to 2 atmospheres while the temperature is kept constant. Find the volume of the air after the pressure is applied. VP = k This is the model for inverse variation. (15)(1) = k Substitute to solve for k.. 15 = k VP = 15 orV = 15 P Find V when P is raised to 2 atmospheres. V = 15 2 V = 7.5 m 3

Joint Variation: Occurs when a quantity varies directly as the product of two or more other quantities. For example, if z = kxy, then z varies jointly with the product of x and y. Example: The variable z varies jointly with the product of x and y. Find an equation that relates the variables if x = 4 when y = –3 and z = 2. z = kxy 2 = k(4)(–3) 2 = –12k k = –1/6 z = –1/6 xy

Example: The Power P in watts of an electrical circuit varies jointly as the resistance R and the square of the current I. For a 600-watt microwave oven that draws a current of 5.0 amperes, the resistance is 24 ohms. What is the resistance of a 200-watt refrigerator that draws a current of 1.7 amperes? P = k R I 2 Joint variation model. 600 = k (24)(5) 2 Substitute to solve for k.. k = k = 1 so P = R I 2 Find R when P is 200 and I is = R(1.7) 2 R = R = 69 ohms